In logic, general frames (or simply frames) are Kripke frames with an additional structure, which are used to model modal and intermediate logics. The general frame semantics combines the main virtues of Kripke semantics and algebraic semantics: it shares the transparent geometrical insight of the former, and robust completeness of the latter.

Definition

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A modal general frame is a triple  , where   is a Kripke frame (i.e.,   is a binary relation on the set  ), and   is a set of subsets of   that is closed under the following:

  • the Boolean operations of (binary) intersection, union, and complement,
  • the operation  , defined by  .

They are thus a special case of fields of sets with additional structure. The purpose of   is to restrict the allowed valuations in the frame: a model   based on the Kripke frame   is admissible in the general frame  , if

  for every propositional variable  .

The closure conditions on   then ensure that   belongs to   for every formula   (not only a variable).

A formula   is valid in  , if   for all admissible valuations  , and all points  . A normal modal logic   is valid in the frame  , if all axioms (or equivalently, all theorems) of   are valid in  . In this case we call   an  -frame.

A Kripke frame   may be identified with a general frame in which all valuations are admissible: i.e.,  , where   denotes the power set of  .

Types of frames

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In full generality, general frames are hardly more than a fancy name for Kripke models; in particular, the correspondence of modal axioms to properties on the accessibility relation is lost. This can be remedied by imposing additional conditions on the set of admissible valuations.

A frame   is called

  • differentiated, if   implies  ,
  • tight, if   implies  ,
  • compact, if every subset of   with the finite intersection property has a non-empty intersection,
  • atomic, if   contains all singletons,
  • refined, if it is differentiated and tight,
  • descriptive, if it is refined and compact.

Kripke frames are refined and atomic. However, infinite Kripke frames are never compact. Every finite differentiated or atomic frame is a Kripke frame.

Descriptive frames are the most important class of frames because of the duality theory (see below). Refined frames are useful as a common generalization of descriptive and Kripke frames.

Operations and morphisms on frames

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Every Kripke model   induces the general frame  , where   is defined as

 

The fundamental truth-preserving operations of generated subframes, p-morphic images, and disjoint unions of Kripke frames have analogues on general frames. A frame   is a generated subframe of a frame  , if the Kripke frame   is a generated subframe of the Kripke frame   (i.e.,   is a subset of   closed upwards under  , and  ), and

 

A p-morphism (or bounded morphism)   is a function from   to   that is a p-morphism of the Kripke frames   and  , and satisfies the additional constraint

  for every  .

The disjoint union of an indexed set of frames  ,  , is the frame  , where   is the disjoint union of  ,   is the union of  , and

 

The refinement of a frame   is a refined frame   defined as follows. We consider the equivalence relation

 

and let   be the set of equivalence classes of  . Then we put

 
 

Completeness

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Unlike Kripke frames, every normal modal logic   is complete with respect to a class of general frames. This is a consequence of the fact that   is complete with respect to a class of Kripke models  : as   is closed under substitution, the general frame induced by   is an  -frame. Moreover, every logic   is complete with respect to a single descriptive frame. Indeed,   is complete with respect to its canonical model, and the general frame induced by the canonical model (called the canonical frame of  ) is descriptive.

Jónsson–Tarski duality

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The Rieger–Nishimura ladder: a 1-universal intuitionistic Kripke frame.
 
Its dual Heyting algebra, the Rieger–Nishimura lattice. It is the free Heyting algebra over 1 generator.

General frames bear close connection to modal algebras. Let   be a general frame. The set   is closed under Boolean operations, therefore it is a subalgebra of the power set Boolean algebra  . It also carries an additional unary operation,  . The combined structure   is a modal algebra, which is called the dual algebra of  , and denoted by  .

In the opposite direction, it is possible to construct the dual frame   to any modal algebra  . The Boolean algebra   has a Stone space, whose underlying set   is the set of all ultrafilters of  . The set   of admissible valuations in   consists of the clopen subsets of  , and the accessibility relation   is defined by

 

for all ultrafilters   and  .

A frame and its dual validate the same formulas; hence the general frame semantics and algebraic semantics are in a sense equivalent. The double dual   of any modal algebra is isomorphic to   itself. This is not true in general for double duals of frames, as the dual of every algebra is descriptive. In fact, a frame   is descriptive if and only if it is isomorphic to its double dual  .

It is also possible to define duals of p-morphisms on one hand, and modal algebra homomorphisms on the other hand. In this way the operators   and   become a pair of contravariant functors between the category of general frames, and the category of modal algebras. These functors provide a duality (called Jónsson–Tarski duality after Bjarni Jónsson and Alfred Tarski) between the categories of descriptive frames, and modal algebras. This is a special case of a more general duality between complex algebras and fields of sets on relational structures.

Intuitionistic frames

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The frame semantics for intuitionistic and intermediate logics can be developed in parallel to the semantics for modal logics. An intuitionistic general frame is a triple  , where   is a partial order on  , and   is a set of upper subsets (cones) of   that contains the empty set, and is closed under

  • intersection and union,
  • the operation  .

Validity and other concepts are then introduced similarly to modal frames, with a few changes necessary to accommodate for the weaker closure properties of the set of admissible valuations. In particular, an intuitionistic frame   is called

  • tight, if   implies  ,
  • compact, if every subset of   with the finite intersection property has a non-empty intersection.

Tight intuitionistic frames are automatically differentiated, hence refined.

The dual of an intuitionistic frame   is the Heyting algebra  . The dual of a Heyting algebra   is the intuitionistic frame  , where   is the set of all prime filters of  , the ordering   is inclusion, and   consists of all subsets of   of the form

 

where  . As in the modal case,   and   are a pair of contravariant functors, which make the category of Heyting algebras dually equivalent to the category of descriptive intuitionistic frames.

It is possible to construct intuitionistic general frames from transitive reflexive modal frames and vice versa, see modal companion.

See also

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References

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  • Alexander Chagrov and Michael Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.
  • Patrick Blackburn, Maarten de Rijke, and Yde Venema, Modal Logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2001.